Excerpt from the book "Arqka Biological Architecture and Sacred Geometry" by Arturo Ponce de León, Ninón Fregoso and Michael Rice Acquire it here

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There are hundreds if not thousands of geometries and derivations of them. Where to start in this sea of possibilities? While it is true that all order is a subjective order, a frame of reference allows us to locate ourselves and know where we come from, where we are and where we are going. It is a map, not the topography, of the geometric universe. And as any map is improvable, questionable and an approach to the indescribable, life itself. After almost two decades of studying the patterns, the forms, the geometries, of seeing and being seen by them, of listening and observing how they conform, the dynamics of space / time, of matter / energy are drawn and blurred. You should understand something: all geometry is sacred and all that is sacred is geometric. Because all geometry can be fractal and all that is sacred can be geometrical.

At first we believed that the golden ratio was more refined than any other proportion. Then we discard that hypothesis. Nature creates life using many more proportions than the golden one. The golden number, 1.618 ... is an asymptote. It is non-existence. It is the demarcation, it is a limit. It’s the vacuum. And in the cosmic breathing, in the process of life and death, of the inhalation and exhalation of the universe, much geometry intervene, chaos and order, simultaneously. It is complex to think about how order and chaos can occur at the same time. However, when we think about it with the integrated hemispheres, it is possible.

Transcending the illusion of orthogonal constructions implies an interior work of integration of the right hemisphere and the left hemisphere of the brain. When both work in unison, the intuitive part and the logical part create a third possibility: fractality. Physically the brain has the shape of a toroid. When the corpus callosum effectively intercommunicates both parts in the center, a larger toroid is nested. That is why the brain is an antenna that if emptied, like the Buddhist maxim, allows morphic resonance fields to be nested in a wider spectrum. It is not only a matter of climbing in the development of intellectual capacities, but of increasing the possibilities of consciousness emotional, intellectual, motor, sexual and spiritual. This gradually shapes the architectural work, the work of design until it manifests naturally.

After many years of using the logical hemisphere and some others of using the intuitive hemisphere, for brief moments we have seen the natural flow of geometric fractal imbrication. And we see it with the bias of our range of sensory perception that configures our own closed universe of apparent stability. That universe in which, we move. Eventually we understood that emptiness only exists as a concept. All is. Everything exists in different degrees of vibration and resonance. As we explained earlier, the so-called void by the scientific community is too full of energy.

In this way we can think of several accommodations on the map to give us an idea of where we are and where to route the design. So many maps have emerged as human beings and possibilities. In European interpretation we have a well-known arrangement of Platonic solids. The tetrahedron is the heart, the fire element. The octahedron the kidneys, air element; the icosahedron the liver, water element; the hexahedron the lungs, earth element and finally the dodecahedron that represents the ether, the akasha.

However, there are other possibilities and accommodations depending on each culture. In the Hindu interpretation of Platonic solids the icosahedron is the first form, the being, the divine essence of everything, represents the father and then comes the mother represented by the dodecahedron, the cosmos. The third that appears is the double tetrahedron where the duality appears, the yin / yang. The fourth to appear is the cube and is understood as the manifestation of everything on Earth. The last one is the octahedron and they say that this is the nucleus of everything, the crystal, the pyramid, both accommodations, the European or the Hindu, are based on concepts, on human thoughts, not on the geometric logic of Nature.

In the vision of arrangement proposed by the geometric analysis of the logic of Nature, that is, of the phenomenological arrangement, we have that Platonic solids are based on contraction. All platonic solids cannot be considered spiritual but material or tending to solidification. Since they are, contracting. We have two toroidal movements’ implosion and explosion. On the implosive or contraction side we find the direction, the pressure, the gravity, that which closes, that creates space, that it becomes visible. While on the expansive or explosive side is the suction, levitation, opening, the unseen.

So the way to get all the platonic solids by contraction is to start with the tetrahedron, because it is the simplest geometric figure possible. And for this we use a geometric process called truncation. And all truncation is a pressure, either by pushing a vertex or by cutting a vertex. Everything that contracts becomes smaller. If we cut the corners to the tetrahedron, we obtain an Archimedean solid, the truncated tetrahedron, a solid in transition. If we continue cutting the corners we get an octahedron. If we cut the corners to the octahedron, we get another solid Archimedean, the cuboctahedron, which is a form of transition. And if I continue cutting the corners it becomes a cube.

The interesting thing is that if we continue cutting the corners of the hexahedron or cube it becomes an octahedron, it is the reason why one is inside the other, they are reciprocal, dual. So until this level we get stuck. How can we get to the icosahedron / dodecahedron? The way to do it is to stop pushing or cutting the corners and you has to push the edges. And if I push the edges of a cube I get a dodecahedron. And to get the icosahedron I have to push the vertices of the dodecahedron. So the geometric sequence by reduction is tetrahedron or fire, octahedron or air, cube or earth, dodecahedron or ether and finally icosahedron or water. This is a conception based on phenomenology, not on concepts or preconceived ideas as correctly pointed out by the geometer and artist Chester.

So far everything is described on the side of the contraction but on the side of the expansion appear new scenarios. We start with the tetrahedron, the fire because it is the smallest volume per area of any shape. To expand a geometric shape you have to take its edge and open it. So that if I open the edges of a tetrahedron I have a seven-sided figure. And I have to open it up to 94.8 degrees, getting a figure, the chestahedro, with 7 faces, 7 points, 12 vertices, and a polygon that is the transition between an octahedron and a hexahedron. And if I see what is formed inside this figure is an octahedron with a tetrahedron on one of its faces. The way to generate expansion is through the dihedrals, the angle formed by two planes. Up to this point all the tetrahedral have the same size. When I open this implicit form further, it becomes a void, and the faces are imploded towards the center. Now for the first time we have suction.

The protocol to do it is through the dihedral, you have to measure that all angles are equal. When I continue to open the figure, the faces begin to swell. So there are two protocols to do these geometric operations. One is that you have to have the correct angle, and the other is that the vertices are the same. If I continue with the same process I get an icosahedron, the symbol of water and a form emerges: a triangular prism that tends to infinity. In the process comes the chestahedron which is the geometry of the ventricle of the human heart.

What this implies in the spiritual world is that both the cube and the dodecahedron do not exist. In the spiritual world we find that everything is triangular, there are no squares or pentagons and they are all based on dihedrals. On the side of the contraction of platonic solids the chestahedron is in the octahedron, it cannot be easily seen but it is there. We can also find it in the icosahedron, in the dodecahedron, cube and tetrahedron. So this 7-sided shape, the Chestahedron, is the skeleton of both the implosive side and the explosive side.